Two faces of symmetry match in gear-like coupled rotary motors

Rotary molecular motors commonly couple components with different rotational symmetries, but how this gearing controls energy transduction remains unclear. Here we show that symmetry can produce a fundamental tradeoff in coupled stochastic barrierless rotary motors: the gear ratio that maximizes output power under tight coupling is also the ratio that requires the strongest coupling to remain locked. Consequently, weakening the coupling can invert the effect of gearing, transforming a power maximum into a local minimum. At strong driving, both the power-optimal and most slip-susceptible ratios approach symmetry match. Numerical solutions of the Fokker-Planck dynamics for rotors with finite energy barriers reproduce these predictions and reveal how the behavior changes when driving strength is fixed per rotation or per subunit. Motivated by the bacterial flagellar motor, these results identify coupling strength as a key determinant of whether matched rotational symmetries enhance or suppress energy transduction, providing a general design principle for coupled molecular motors.